In a bowl, combine the sourdough discard, milk, mayonnaise, and beaten egg.
In a separate bowl, whisk together the flour, salt, baking powder, and sugar.
Gradually add the dry ingredients to the wet ingredients, stirring until combined.
Add the melted butter and mix until the dough comes together.
Turn the dough out onto a floured surface and knead for 5 minutes, or until smooth and elastic.
Shape the dough into 6 equal-sized balls and place them on a baking sheet lined with parchment paper.
Cover the dough balls with a clean kitchen towel and let them rise for 1-2 hours, or until doubled in size.
Preheat the oven to 375°F (190°C).
Brush the risen dough balls with milk and sprinkle sesame seeds on top.
Bake for 15-18 minutes, or until golden brown.
Let the hamburger buns cool on a wire rack before slicing and serving.
